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Flower garden sequence (see Comments for precise definition).
10

%I #135 Jun 18 2022 18:22:07

%S 0,1,3,7,15,19,27,39,63,67,75,87,111,123,147,183,255,259,267,279,303,

%T 315,339,375,447,459,483,519,591,627,699,807,1023,1027,1035,1047,1071,

%U 1083,1107,1143,1215,1227,1251,1287,1359,1395,1467,1575,1791,1803,1827,1863,1935,1971,2043,2151,2367,2403,2475

%N Flower garden sequence (see Comments for precise definition).

%C This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks and V-toothpicks. Also, it appears that this is a missing link between A147562 (Ulam-Warburton) and three toothpick sequences: A139250 (normal toothpicks), A161206 (V-toothpicks) and A160120 (Y-toothpicks). The behavior resembles the toothpick sequence A139250, on the other hand, the formulas are directly related to A147562. Plot 2 shows that the graph is located between the graph of A139250 and the graph of A147562.

%C For the construction of the sequence the rules are as follows:

%C On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.

%C At stage 1 we place an I-toothpick formed of two single toothpicks in vertical position, so a(1) = 1.

%C For the next n generation we have that:

%C If n is even then at every free end of the structure we add a V-toothpick formed of two single toothpicks such that the angle of each single toothpick with respect to the connected I-toothpick is 120 degrees.

%C If n is odd then we add I-toothpicks in vertical position (see the example).

%C a(n) gives the total number of I-toothpicks and V-toothpicks in the structure after the n-th stage.

%C A323651 (the first differences) gives the number of elements added at the n-th stage.

%C Note that 2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.

%C The structure contains only three kinds of polygonal regions as follows:

%C - Rhombuses that contain two triangular cells.

%C - Regular hexagons that contain six triangular cells.

%C - Oblong hexagons that contain 10 triangular cells.

%C The structure looks like a "garden of flowers with six petals" (between other substructures). In particular, after 2^(n+1) stages with n >= 0, the structure looks like a flower garden in a rectangular box which contains A002450(n) flowers with six petals.

%C Note that this hybrid cellular automaton is also a superstructure of the Ulam-Warburton cellular automaton (at least in four ways). The explanation is as follows:

%C 1) A147562(n) equals the total number of I-toothpicks in the structure after 2*n - 1 stage, n >= 1.

%C 2) A147562(n) equals the total number of pairs of Y-toothpicks connected by their endpoints in the structure after 2*n stage (see the example).

%C 3) A147562(n) equals the total number of "flowers with six petals" (or six-pointed stars formed of six rhombuses) in the structure after 4*n stage. Note that the location of the "flowers with six petals" in the structure is essentially the same as the location of the "ON" cells in the version "one-step bishop" of A147562.

%C 4) For more connections to A147562 see the Formula section.

%C The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

%C The total number of “flowers with six petals” after n-th stage equals the total number of “hidden crosses” after n-th stage in the toothpick structure of A139250, including the central cross (beginning to count the crosses when their “nuclei” are totally formed with 4 quadrilaterals). - _Omar E. Pol_, Mar 06 2019

%H Paolo Xausa, <a href="/A323650/b323650.txt">Table of n, a(n) for n = 0..9999</a>

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>

%F a(n) = 3*A147562(n/2) if n is even.

%F a(n) = 3*A147562((n-1)/2) + A147582(n) if n is odd.

%F a(n) = 3*A147562((n-1)/2) + A147562(n) - A147562(n-1) if n is odd.

%F a(2^n) = A103454(n), n >= 0.

%e Illustration of initial terms:

%e .

%e | |

%e \ / |\ /|

%e | | |

%e | | |

%e / \ |/ \|

%e | |

%e n 1 2 3

%e a(n) 1 3 7

%e .

%e Note that for n = 2 the structure is also the same as a pair of Y-toothpicks connected by their endpoints (see A160120).

%t bw3[n_]:=bw3[n]=3^(DigitCount[n,2,1]-1);

%t A147562[n_]:=A147562[n]=If[n<2,n,1+4Sum[bw3[k],{k,n-1}]];

%t A323650[n_]:=If[OddQ[n],3A147562[(n-1)/2]+A147562[n]-A147562[n-1],3A147562[n/2]];

%t nterms=100;Array[A323650,nterms,0] (* _Paolo Xausa_, Jun 17 2022 *)

%Y Cf. A002450, A103454, A139250 (normal toothpicks), A147562 (Ulam-Warburton), A147582, A160120 (Y-toothpicks), A161206 (V-toothpicks), A296612, A323641, A323642, A323649 (corner sequence), A323651 (first differences).

%Y For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770.

%K nonn

%O 0,3

%A _Omar E. Pol_, Jan 21 2019